Page 40 - Volume 9, Issue 3

THE WAVEGUIDE INVARIANT:
SPATIAL INTERFERENCE PATTERNS IN UNDERWATER ACOUSTICS
Chris H. Harrison
Institute of Sound and Vibration Research University of Southampton
Highfield, Southampton SO17 1BJ
United Kingdom
and
Centre for Maritime Research and Experimentation Viale San Bartolomeo 400
19126 La Spezia
Italy
Introduction
“If you see some fringes,
images and the actual receiver. For the time being assuming the sound speed to be constant, it is clear that the earliest arrivals are bunched up, but as time pro- gresses the later arrivals get further and further apart. By analogy with the
In long range underwater acoustics
there is a simple quantity known as
the waveguide invariant that tells
you how acoustic spatial interference
patterns change as you move a distant
receiver horizontally away from a sound source. More than just a curiosity, this quantity is beginning to find naval applications. A particularly attractive point is that interfer- ence fringes or striations enable location or ranging of a dis- tant sound source using just a single hydrophone. But before I say exactly how it works, there are three other apparently disparate phenomena which are closely related, and this relationship can be understood by thinking about them all in the time domain. Furthermore they help under- stand what the fringes depend on.
The first phenomenon I noticed when I was a teenager with “clip-clop-shoes” walking down a quiet paved street next to a feather-edged fence, the type with vertical overlap- ping planks. The surprise was that the steel-tipped heels no longer went clip, clop. Instead they went “pyank, pyonk... ”. In other words the original click or delta function was turned into a rapid downward frequency sweep.
The second phenomenon is the well known similar effect when you clap your hands near the steps of the Mexican zig- gurat of Chichen Itza (see Declercq, 2013). The echo is a also a downward frequency sweep.
While doing some noise experiments in the Mediterranean with underwater sound being monitored by loudspeakers in the ship’s lab I witnessed the third phenom- enon. This was a randomly occurring, isolated rather strange sound that was rather like a Walt Disney Goofy’s gulp. You can reproduce this sound by trying to say “Eeyore” like a donkey but with your mouth shut! Incidentally it sounds much louder with your head under water! The similarity is that it is also a downward sweep but at a much lower starting frequency.
The cause of these sounds was allegedly distant oil- prospecting air guns heard through a multipath environment where rays zig-zag many times between sea surface and bot- tom before reaching the receiver. The impulse response of a single distant shot can be revealed by unravelling the zig-zags to make straight paths between a vertical array of source
don’t throw them away,
they could be useful.”
pitched buzz of a playing card on bicycle wheel spokes, this sequence of ever more separated clicks sounds like a rather buzzy downward frequency sweep. What’s in common with all three phenomena is the effective row of sources of clicks, all more or less at right-angles to the main path. We can replace the column of image sources with the row of reflect- ing corners in the feather-edged fence or in the Chichen Itza steps.
More formally, the height difference between the receiv- er and source image is 2nH + μzs + vzr where n is an integer, H is the water depth, zs and zr are the true source and receiv- er depths, and μ and v both take values of +1 and –1. So with sound speed c at range r in the small angle approximation the travel time delay after the first return tn,μ,v is
(1)
The quadratic relationship between n and t is responsi- ble for the early bunching up of the impulses. If you don’t have access to bicycle spokes or playing cards you can do just as well in Matlab by setting up an array of zeros at some sam- pling frequency and then overwriting with a one at each of the delays indicated above and listening to this irregularly spaced “row of spikes” with the function ‘sound’. Subtle dif- ferences may be introduced into the nature of the sound by altering the detailed positions of the spikes through the source and receiver depths and also by altering their ampli- tudes according to reflection coefficients, angles, and so on. Figure 1 shows a typical impulse response generated this way.
The waveguide invariant b
This simple-minded ray explanation seems perfectly nat-
ural at audio frequencies in the air acoustics case, but it may come as more of a shock to physicists and mathematicians who work in underwater acoustics. Typically they are taught to find the relevant differential equation, choose the bound- ary conditions for the particular problem, and solve it some-
The Waveguide Invariant 39